Five points are located on a certain straight line
Nine pairwise distances are known:
2,4,5,7,12,13,15,17,19
Find the 10th.
Based on Hoshino & McCurdy puzzle published in Crux Mathematicorum
A pair of adjacent shorter segments will add up to a larger segment. So that suggests trying to form addition equations from the nine numbers given:
2+5=7
2+13=15
2+15=17
2+17=19
4+13=17
4+15=19
5+7=12
5+12=17
7+12=19
Exactly four of these numbers occur only as addends on the left: 2, 4, 5, 13. It is possible for these four numbers to be the lengths of the four segments between consecutive points on the line. However this is a problem, as there is no subset of 2,4,5,13 which sums to 12.
The next possibility is that one of the four single segments to be the number that was deleted, with 13 being the sum of two single segments. In this case this also means that 19 is the total length of the line. 2+4+13=19, leaving out 5. Then 13-5=8 is the missing segment.
To verify, construct the line: the line is comprised of single segments of 2,4,5,8 in some order with 2,4,5,7,8,12,13,15,17,19 being all the pairwise distances.
2+4=6 and 2+7=9 are not given distances, so 2 is at an end adjacent to 5. 5+4=9 is also not given so 4 is not adjacent to 5.
This is enough to determine the order is 2, 5, 8, 4. Then the 10 lengths in order are 2, 4, 5, 2+5=7, 8, 8+4=12, 5+8=13, 2+5+8=15, 5+8+4=17, 2+5+8+4=19.
Analytic Solution
